This proof adapts
the technique of the proof of [1]}'s Proposition
1. Without loss of generality we may assume that \({\displaystyle \int _{\mathbb {R}^{p}}\exp (-U(x))dx=1}\) .
We first give upper and lower bounds to the normalizing constant of
\(\pi _{\mu }\) , that is
\(c_{\mu } & \stackrel{_{\triangle }}{=}\int _{\mathbb {R}^{d}}\pi (x)e^{-\left(U_{\mu }(x)-U(x)\right)}dx.\\& =\mathbb {E}_{\pi }\left(e^{-\left(U_{\mu }(x)-U(x)\right)}\right)\) 
